Question: Simplify the following expression and state the condition under which the simplification is valid: $q = \dfrac{t^2 + 3t - 54}{t^2 - 6t}$
First factor the expressions in the numerator and denominator. $ \dfrac{t^2 + 3t - 54}{t^2 - 6t} = \dfrac{(t + 9)(t - 6)}{(t)(t - 6)} $ Notice that the term $(t - 6)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(t - 6)$ gives: $q = \dfrac{t + 9}{t}$ Since we divided by $(t - 6)$, $t \neq 6$. $q = \dfrac{t + 9}{t}; \space t \neq 6$